Question: Mr. Mendez awards extra credit on quizzes to his students with quiz grades that exceed the class mean. Given that 107 students take the same quiz, what is the largest number of students who can be awarded extra credit?
Solution: It is not possible for all 107 to exceed the mean since the mean is always greater than or equal to the smallest element.  However 106 of the students can exceed the mean.  For example, if 106 students get a 5 and the other student gets a 4, the mean is slightly less than 5 and all $\boxed{106}$ of the students that scored 5 will exceed the mean.